## Abstract

This paper presents an evaluation of derived dewpoint temperature and derived relative humidity, in which the dewpoint temperature is calculated using measured ambient air temperature and measured relative humidity variables and the derived relative humidity is calculated from measured dewpoint temperature. The derived dewpoint temperature and relative humidity are calculated using algorithms provided by the World Meteorological Organization. The method of uncertainty analysis, provided by the National Institute of Standards and Technology, is applied to calculate the uncertainties of an indirect measurement of derived dewpoint temperature and derived relative humidity. The results from the uncertainty analyses of derived and observed variables suggest that the use of derived dewpoint temperature and derived relative humidity involves risk because the uncertainties of modern dewpoint temperature and relative humidity sensors can create several degrees Celsius of error in the derived dewpoint temperature and several percent in the derived relative humidity.

## Introduction

A single air humidity sensor or system usually is capable of directly measuring a single humidity-related variable such as relative humidity by use of a solid-state humidity sensor or dewpoint temperature from a chilled mirror hygrometer. Use of these technologies is common with weather-station networks or controlled environmental systems. However, in many agricultural and engineering applications, the dewpoint temperature could be the required variable when the relative humidity was measured, or vice versa. In such cases, we usually perform some calculations to derive the desired humidity variables from existing data and available algorithms. The question for us is, How accurate are these derived humidity variables and how can we quantify their uncertainties?

The concept of the uncertainty of measurements was newly defined in a set of guidelines by the National Institute of Standards and Technology (NIST 1994) that is a technical summary of a longer International Organization for Standardization guide (ISO 1993), which was updated and corrected in 1995. The uncertainty is a parameter, associated with the result of a measurement, that characterizes the dispersion of the values that reasonably could be attributed to the measurand. This concept has been internationally accepted as the basis for the determination of the measurement uncertainty. Application of these guidelines has extended beyond calibration and research laboratories and into the industrial domain of manufactured products. Our intent is to obtain the uncertainties of derived dewpoint temperature and derived relative humidity when derived using the calculation algorithms provided by the World Meteorological Organization (WMO 1996).

## Method

More than a dozen simple or complicated empirical equations exist for calculating the saturation vapor pressure from ambient temperature and ambient air pressure or calculating the actual vapor pressure from the dewpoint temperature based on the Goff–Gratch (1946) formulation. The equations that are commonly used in atmospheric sciences and other related disciplines are from Wexler (1976), Buck (1981), and WMO (1996). In this study, the WMO equations were taken for the uncertainty analysis as follows (WMO 1996):

where *e*_{ws} is saturation vapor pressure (hPa) with respect to water at the air temperature *T*_{a} (°C), *f*(*P*) [=1.0016 + (3.15 × 10^{−6})*P* − 0.0074*P*^{−1}] is the enhancement factor at the air pressure *P* (hPa), RH is relative humidity (%), and *e*_{w} is actual vapor pressure (hPa).

The saturation vapor pressure *e*_{ws} is a function of air temperature and site atmospheric pressure, whereas the actual vapor pressure *e*_{w} is a function of dewpoint temperature *T*_{d} and site atmospheric pressure if the saturation vapor pressure is with respect to the water or is a function of frostpoint temperature if the saturation vapor pressure is with respect to ice. Thus, the WMO Eqs. (1) and (2) are often used to calculate the derived *T*_{d} from known values of RH and *T*_{a} and to calculate the derived RH from known values of the *T*_{d} and *T*_{a}. The calculations for both RH and *T*_{d} obviously require the measurements of site atmospheric pressure. In our uncertainty analysis, the site or station atmospheric pressure is considered to be a constant (1000 hPa).

In uncertainty analysis, the individual standard uncertainty *u*_{i} is defined as the uncertainty of the result of a measurement expressed as its standard deviation (NIST 1994). For the manufacturer's stated “accuracy” specifications of air temperature, relative humidity, and dewpoint temperature sensors, it is reasonable to use rectangular distributions in uncertainty analysis (NIST 1994). Thus, one must assume a rectangular distribution for all sensors that measure *T*_{a}, RH, and *T*_{d} based on the manufacturer specifications with a lower limit *a*_{−} and an upper limit *a*_{+} (e.g., ±0.2°C for *T*_{a} or *T*_{d} and ±2% for RH). Therefore, the standard uncertainty (NIST 1994) of *T*_{d} and RH is

where *a* = (*a*_{+} − *a*_{−})/2 for both dewpoint temperature and relative humidity, and *x*_{i} represents *T*_{a}, *T*_{d}, and RH. The combined standard uncertainty *u*_{c} can be derived from the original Eqs. (1) and (2) using the law of propagation of uncertainty, commonly called the root sum-of-squares (RSS) method for a specific output quantity (*T*_{d} or RH). Multiplying a coverage factor *k* (NIST 1994) by the combined standard uncertainty, one can obtain the expanded uncertainty *U.* In accord, a normal coverage distribution will be used for expressing uncertainty in this study and is defined by setting *k* = 2 such that the uncertainty interval for *U* has a 95% confidence level. We assume that, in each system, the measurement of dewpoint temperature or the measurement of relative humidity is statistically independent of air temperature measured in observations. Therefore, the expanded uncertainties for derived dewpoint temperature *U*_{Td} and derived relative humidity *U*_{RH} in each system can be calculated as

The partial derivatives in Eq. (4) are often referred to as sensitivity coefficients and can be obtained analytically. From Eq. (3), *u*(*T*_{a}), *u*(*T*_{d}), and *u*(RH) are the standard uncertainty associated with the manufacturer's specifications for each sensor. The value of *k* is 2 as described above. To illustrate the uncertainty of derived dewpoint temperature and derived relative humidity, we assume that the accuracy specifications from the manufacturers are ±0.2° or ±0.3°C for air temperature, ±2% or ±5% for relative humidity, and ±0.3° or ±0.5°C for dewpoint temperature. For the derived dewpoint temperature calculations, we took the known ambient temperature for a range extending from −50° to +50°C and the known ambient relative humidity extending across a range from 2% to 100%. For derived relative humidity calculations, the ambient temperature range was set between −50° and +50°C and the dewpoint temperature depression (difference between ambient air temperature and dewpoint temperature) was limited to a range of 0°–50°C.

## Results and discussion

### Uncertainty of derived dewpoint temperature

Figure 1 illustrates the variation of the uncertainty of derived dewpoint temperature with ambient temperature and relative humidity. The uncertainty of derived dewpoint temperature (*U*_{Td}) increases as the known relative humidity decreases. With decreasing ambient air temperature, the uncertainty of derived dewpoint temperature decreases. When the known relative humidity is 10%, the uncertainty of derived dewpoint temperature ranges from 1.8° to 3.3°C, resulting in a system with an air temperature accuracy of ±0.2°C and a relative humidity accuracy of ±2% (Table 1). From Fig. 1 and Table 1, in principle, for any dewpoint temperature instrument or sensor based upon the relative humidity measurement, the uncertainty of dewpoint temperature measurement cannot fall below ±0.5°C unless the manufacturer uses a more accurate saturation vapor pressure equation than that provided by the WMO in Eq. (1). Furthermore, if the known ambient temperature and relative humidity have an accuracy of ±0.3°C and ±5%, respectively, this combination creates a range of uncertainty for derived dewpoint temperature of 0.7°– 8.1°C (Table 1). In reality, the accuracy combination of ±0.3°C and ±5% for the ambient temperature and relative humidity is a realistic expectation when quality sensors are deployed in remote field conditions.

### Uncertainty of derived relative humidity

The uncertainty of derived relative humidity is illustrated in Fig. 2. In general, the uncertainty of derived relative humidity exponentially increases with a decreasing dewpoint temperature depression or an increasing relative humidity. When the dewpoint temperature depression is relatively small, increasing the ambient temperature decreases the uncertainty of derived relative humidity (Fig. 2 and Table 2). Under the accuracy combination of ±0.2° and ±0.3°C for ambient air temperature and dewpoint temperature depression, the uncertainty of derived relative humidity ranges from 0.4% to 6.6% (Table 2). The uncertainty of derived relative humidity increases when an accuracy combination of ±0.3° and ±0.5°C is used for ambient air temperature and dewpoint temperature depression (Fig. 2). In a similar way, the latter combination of measurements is a more reasonable expectation for users using modern sensors and practices under field conditions. Thus, the uncertainty of derived relative humidity should range from 0.6% to 10.9% (Table 2 and Fig. 2).

The results in Figs. 1 and 2 reflect the imperfections in the WMO equation under certain conditions when humidity variables are derived. However, even when other equations such as the Wexler (1976) equation or American Society of Heating, Refrigeration, and Air Conditioning Engineers (ASHRAE 1997) equation were used to calculate saturation vapor pressure, similar results were produced. In addition, Gates (1994) presented the expected error in derived dewpoint temperature using a measured dry-bulb temperature and relative humidity and the ASHRAE equation. Note that the expected error or root-mean-square error used in Gates (1994) indicates one standard deviation in the units of measure (a 68% confidence level for normal distribution). However, the method used by Gates to simulate the results was based on the RSS rather than the expected error or the root-mean-square error (Gates 1994). The results presented in our study reflect the uncertainties of derived dewpoint temperature and derived relative humidity at the 95% confidence level. To the authors' knowledge, the accuracy combinations shown in the third column of Tables 1 and 2 (top panel of Figs. 1 and 2) are achievable in the laboratory, but the results illustrated in the fourth column are more realistic of the observations from remotely deployed sensors (bottom panel of Figs. 1 and 2).

## Conclusions

Significant uncertainties exist when dewpoint temperature and relative humidity are derived using data from automated weather stations, climate databases that include derived humidity variables (*T*_{d} or RH), and humidity sensors designed to measure one variable and programmed to output another. The magnitude of the uncertainty for derived dewpoint temperature not only increases with increasing ambient temperature, but also increases with the falling ambient relative humidity. The magnitude of uncertainty for derived relative humidity increases with decreasing dewpoint temperature depression. It also changes with ambient temperature, especially when ambient relative humidity is high, which in reality is a common condition. We recommend that one should use, when possible, the actual direct measurements for air humidity–related observation. However, if derived air humidity data must be used, the uncertainty should be calculated, especially for derived dewpoint temperatures when the relative humidity is low and for derived relative humidity when the relative humidity is high. The hope for a reduction in uncertainty of derived humidity variables lies in examining the bias relationship between measured humidity variables and other variables (e.g., ambient temperature of sensors and ambient wind speed). Transfer functions that remove this bias will reduce the input uncertainty in the uncertainty analysis and will result in lower uncertainty of the derived variables. We intend to study this aspect more in the future.

## Acknowledgments

This study was supported in part by the National Climatic Data Center (NCDC) in our project of “Performance Study of Air Humidity/Water Vapor Monitoring Systems for the U.S. Climate Reference Network (USCRN).” We thank Nebraska State Climatologist Allen L. Dutcher and Dr. Qi Hu for their valuable comments and suggestions on the original manuscript. This paper has been approved as Journal Series No. 14227 by the Agricultural Research Division, University of Nebraska at Lincoln.

## REFERENCES

## Footnotes

*Corresponding author address:* Dr. Kenneth G. Hubbard, 244 L. W. Chase Hall, School of Natural Resource Sciences, University of Nebraska—Lincoln, Lincoln, NE 68583-0728. khubbard1@unl.edu